We formulate a formal grammar to generate Full Approximation Scheme multigrid solvers. Then, using Grammar-Guided Genetic Programming we perform a multiobjective optimization to find optimal instances of such solvers ...
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(纸本)9798400701207
We formulate a formal grammar to generate Full Approximation Scheme multigrid solvers. Then, using Grammar-Guided Genetic Programming we perform a multiobjective optimization to find optimal instances of such solvers for a given nonlinear system of equations. This approach is evaluated for a two-dimensional Poisson problem with added nonlinearities. We observe that the evolved solvers outperform the baseline methods by having a faster runtime and a higher convergence rate.
We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture mo...
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We develop a thermodynamically consistent mixture model for avascular solid tumor growth which takes into account the effects of cell-to-cell adhesion, and taxis inducing chemical and molecular species. The mixture model is well-posed and the governing equations are of Cahn-Hilliard type. When there are only two phases, our asymptotic analysis shows that earlier single-phase models may be recovered as limiting cases of a two-phase model. To solve the governing equations, we develop a numerical algorithm based on an adaptive Cartesian block-structured mesh refinement scheme. A centered-difference approximation is used for the space discretization so that the scheme is second order accurate in space. An implicit discretization in time is used which results in nonlinear equations at implicit time levels. We further employ a gradient stable discretization scheme so that the nonlinear equations are solvable for very large time steps. To solve those equations we use a nonlinear multilevel/multigrid method which is of an optimal order O(N) where N is the number of grid points. Spherically symmetric and fully two dimensional nonlinear numerical simulations are performed. We investigate tumor evolution in nutrient-rich and nutrient-poor tissues. A number of important results have been uncovered. For example, we demonstrate that the tumor may suffer from taxis-driven fingering instabilities which are most dramatic when cell proliferation is low, as predicted by linear stability theory. This is also observed in experiments. This work shows that taxis may play a role in tumor invasion and that when nutrient plays the role of a chemoattractant, the diffusional instability is exacerbated by nutrient gradients. Accordingly, we believe this model is capable of describing complex invasive patterns observed in experiments.
We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn-Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently...
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We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn-Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and the anisotropic Cahn-Hilliard equation becomes ill-posed. To regularize the equation, a higher-order derivative term is added to the energy. This leads to a sixth-order, nonlinear parabolic equation for the order parameter. An implicit time discretization is used to remove the high-order time step stability constraints. Dynamic block-structured Cartesian mesh refinement is used to highly resolve narrow interfacial layers. A multilevel, nonlinearmultigrid method is used to solve the nonlinear equations at the implicit time level. One of the keys to the success of the method is the treatment of the anisotropic term. This term is discretized in conservation form in space and is discretized fully implicitly in time. Numerical simulations are presented that confirm the accuracy, efficiency and stability of the scheme. We study the dynamics of interfaces under strong anisotropy and compare near-equilibrium diffuse interface solutions to the sharp interface Wulff shapes in 2D and 3D. We also simulate large-scale coarsening of a corrugated surface (in 3D) evolving by anisotropic surface diffusion. We show the emergence of long-range order during coarsening and an interesting mechanism of ordered coarsening. (C) 2007 Elsevier Inc. All rights reserved.
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